where \(\theta _\mathrm {min}=0\)  for
the pole (\(\theta _\mathrm {min}\ne 0\)  for a ring; see e.g. [1]}) and \(\theta _\mathrm {max}\leqslant 90{\mbox{°}}\)  is the equivalent opening angle of the
accreting region
at each pole. For \(f_{\mathrm {fill}}=0.3\)  (e.g. 15 % of the total area at the north pole and 15 % at the south pole), the equivalent opening angle from \(\theta _\mathrm {min}=0{\mbox{°}}\)  is \(\theta _\mathrm {max}=46{\mbox{°}}\) .
For a ring with \(f_{\mathrm {fill}}=0.1\)  starting and at \(\theta _\mathrm {min}=20{\mbox{°}}\)  (for example), \(\theta _\mathrm {max}=50{\mbox{°}}\) , while \(f_{\mathrm {fill}}=0.01\)  would require \(\theta _\mathrm {max}=21.6{\mbox{°}}\) , or \(\theta _\mathrm {max}=12.9{\mbox{°}}\)  if instead \(\theta _\mathrm {min}=10{\mbox{°}}\) .
